Q1) Write in the following as a single cycle or a product of disjoint cycles?

A) ( 1 2 3 ) ^(-1) ( 2 4 ) ( 1 2 3 )

I do this first one. First by the other post. Thus, we need to simplify remember this means . To do these problems we map and then find then until we complete the cycle. Note, by applying first then by applying second and by applying third. Thus, by . Now we find . Note by first then under and finally by . Thus, and it means we only write as we express as a product of disjoint cycles. Next we do the same with . See that under and so . Which means the representation is .